Every unlocked achievements gives a +5 bonus to all attributes. If you have 18/50, you get a total bonus of +90 to all attributes for example.
Some of the achievements below are easy, some are for high level-players and some are crazy.
Sometimes an achievement that requires items does not unlock. Unequip an item and equip it again to solve this issue.

Naturism: be 24 hours without equipment but in one go! Hint: play as many quests on day 1, then unequip your items. Then equip them again on day 2 and use the remaining hours to do quests. This way, you do not lose gold and XP from quests.

Ri-off Rip-off: beat the Gambler 3 times in a row, either with gold or mushrooms

Heavy Spinner: spin the Wheel 20 times a day

Square Eyes: collect 1,000 Lucky Coins

Lucky Fellow: find 100 mushrooms on quests

Raider: raid 100 fortresses

Invincible: win the first 10 arena fights of the day (the ones you get XP for)

Song of the Blacksmith: hoard 10,000 arcane resources

Sky is the Limit: improve an item 10 times

Miner: mine 100 things

Collecting Mania: scrapbook 90% complete

Anniversary: play for 1 year

For Seven Days: play 7 days in a row

Always On: play 30 days in a row

Reliable: be actively involved in a guild for 10 days...by taking part in guild attacks and defenses and fighting in the guild portal

Elite Guild: reach 5,000 Guild Honor

Big Spender: donate 1,000,000 gold to guild. If your guild bank is full, you need to leave the guild and donate elsewhere. It's the only solution at the moment.

Hero: reach level 100

Elite: reach level 200

Tip of the Iceberg: reach level 300

Superhero: reach level 400

Epic Superhero: reach level 500

Jeweler: have gems in all equipped items

Black Gold: wear only items with black gems

Fashion-conscious: wear 6 items of a set, check out the second post in this thread for the sets of each class

Please note that the name of an item is not that important. "Chieftain's Shield" does not automatically go with "Chieftain's Boots" and "Chieftain's Gloves" but some sets consist of items with mixed names. Example: